$$\DeclareMathOperator{cosec}{cosec}$$

Statistics and Probability

Syllabus Content

Variance of a discrete random variable.
Continuous random variables and their probability density functions.
Mode and median of continuous random variables.
Mean, variance and standard deviation of both discrete and continuous random variables.
The effect of linear transformations of X.

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Official Guidance, clarification and syllabus links:

Link to: discrete random variables (SL 4.7)

$$0 \le f(x) \le 1,\int_{-\infty}^{\infty} f(x) dx=1$$ including piecewise functions.

For a continuous random variable, a value at which the probability density function has a maximum value is called a mode and for the median:

$$\int_{-\infty}^{m} f(x) dx= \frac{1}{2}$$

Use of the notation $$E(X), E(X^2),Var(X), \text{ where } Var(X)=E(X^2)-[E(X)]^2$$ and related formulae.

Use of $$E(X)$$ for "fair" games.

$$E(aX+b)=aE(X)+b, Var(aX+b)=a^2Var(X)$$

Formula Booklet:

The variance of a discrete random variable measures the spread or dispersion of the probability distribution. Mathematically, if $$X$$ is a discrete random variable with mean $$\mu$$ and probability mass function $$p(x)$$, the variance $$\sigma^2$$ is given by: $$\sigma^2 = \sum (x - \mu)^2 \cdot p(x)$$

For continuous random variables, we use probability density functions (pdfs) instead of probability mass functions. The pdf, denoted by $$f(x)$$, gives the likelihood of the variable taking on a particular value. The total area under the curve of a pdf is always 1. The mode of a continuous random variable is the value of $$x$$ for which the pdf has its maximum value. The median is the value $$m$$ such that the area under the curve to the left of $$m$$ is 0.5.

The mean (or expected value) of both discrete and continuous random variables is a measure of the central tendency. For a discrete random variable $$X$$ with probability mass function $$p(x)$$, the mean $$\mu$$ is: $$\mu = \sum x \cdot p(x)$$ For a continuous random variable with pdf $$f(x)$$, the mean is: $$\mu = \int_{-\infty}^{\infty} x \cdot f(x) \, dx$$ The variance and standard deviation for continuous random variables can be found using similar formulas, but with integration instead of summation.

When we apply a linear transformation of the form $$Y = aX + b$$ to a random variable $$X$$, the mean and variance change in predictable ways. The new mean $$\mu_Y$$ and variance $$\sigma_Y^2$$ are given by: $$\mu_Y = a\mu_X + b \\ \sigma_Y^2 = a^2\sigma_X^2$$ The standard deviation $$\sigma_Y$$ of $$Y$$ is $$|a|$$ times the standard deviation $$\sigma_X$$ of $$X$$.

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